2008 iTest Problems/Problem 78

Problem

Feeling excited over her successful explorations into Pascal's Triangle, Wendy formulates a second problem to use during a future Jupiter Falls High School Math Meet:

How many of the first 2010 rows of Pascal's Triangle (Rows 0 through 2009) have exactly 256 odd entries?

What is the solution to Wendy's second problem?

Solution

Building off of Method 2 in problem 77, we see that each that the base 2 representation of the number reveals to us how many times we end up applying the recursion relation $O_k=2*O_{k-2^{n-1}}$ .

For example, $2008=1024+512+256+128+64+16+8=(11111011000)_2$ has seven nonzero terms in its base 2 expansion so that we end up applying the recursion relation seven times giving a final term of $O_{2008}=2^7\cdot O_{0}=2^7=128$ . So we need to count the number of integers between $\{0,2009\}$ which have exactly 8 nonzero terms in their base 2 expansion.

$2009=(11111011001)_2$ .

$\binom{11}{8}$ counts the numbers up to $2047=2^{11}-1$ which have exactly 8 nonzero terms in their base 2 expansion. Of these $\binom{11}{8}=165$ candidates, the following 12 must be excluded since they exceed $2009$ :